Find the square root of the following numbers using the Prime Factorisation Method: (i) 729 (ii) 400 (iii) 1764 (iv) 4096

Question

Asked on April 16, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-04-16T13:33:15.601Z

You're on a roll — Step 1: Find the square root of each number using the Prime Factorisation Method. (i)* 729 Find the prime factors of 729: 729 = 3 × 243 243 = 3 × 81 81 = 3 × 27 27 = 3 × 9 9 = 3 × 3 So, 729 = 3 × 3 × 3 × 3 × 3 × 3 = (3 × 3 × 3)^2 = 27^2. sqrt(729) = sqrt(3^6) = 3^6/2 = 3^3 = 27 (ii)* 400 Find the prime factors of 400: 400 = 2 × 200 200 = 2 × 100 100 = 2 × 50 50 = 2 × 25 25 = 5 × 5 So, 400 = 2 × 2 × 2 × 2 × 5 × 5 = (2 × 2 × 5)^2 = 20^2. sqrt(400) = sqrt(2^4 × 5^2) = 2^4/2 × 5^2/2 = 2^2 × 5^1 = 4 × 5 = 20 (iii)* 1764 Find the prime factors of 1764: 1764 = 2 × 882 882 = 2 × 441 441 = 3 × 147 147 = 3 × 49 49 = 7 × 7 So, 1764 = 2 × 2 × 3 × 3 × 7 × 7 = (2 × 3 × 7)^2 = 42^2. sqrt(1764) = sqrt(2^2 × 3^2 × 7^2) = 2^1 × 3^1 × 7^1 = 2 × 3 × 7 = 42 (iv)* 4096 Find the prime factors of 4096: 4096 = 2 × 2048 2048 = 2 × 1024 1024 = 2 × 512 512 = 2 × 256 256 = 2 × 128 128 = 2 × 64 64 = 2 × 32 32 = 2 × 16 16 = 2 × 8 8 = 2 × 4 4 = 2 × 2 So, 4096 = 2^12 = (2^6)^2 = 64^2. sqrt(4096) = sqrt(2^12) = 2^12/2 = 2^6 = 64 (v)* 7744 Find the prime factors of 7744: 7744 = 2 × 3872 3872 = 2 × 1936 1936 = 2 × 968 968 = 2 × 484 484 = 2 × 242 242 = 2 × 121 121 = 11 × 11 So, 7744 = 2 × 2 × 2 × 2 × 2 × 2 × 11 × 11 = (2 × 2 × 2 × 11)^2 = (8 × 11)^2 = 88^2. sqrt(7744) = sqrt(2^6 × 11^2) = 2^6/2 × 11^2/2 = 2^3 × 11^1 = 8 × 11 = 88 (vi)* 9604 Find the prime factors of 9604: 9604 = 2 × 4802 4802 = 2 × 2401 2401 = 7 × 343 343 = 7 × 49 49 = 7 × 7 So, 9604 = 2 × 2 × 7 × 7 × 7 × 7 = (2 × 7 × 7)^2 = (2 × 49)^2 = 98^2. sqrt(9604) = sqrt(2^2 × 7^4) = 2^2/2 × 7^4/2 = 2^1 × 7^2 = 2 × 49 = 98 (vii)* 5929 Find the prime factors of 5929: 5929 = 7 × 847 847 = 7 × 121 121 = 11 × 11 So, 5929 = 7 × 7 × 11 × 11 = (7 × 11)^2 = 77^2. sqrt(5929) = sqrt(7^2 × 11^2) = 7^1 × 11^1 = 7 × 11 = 77 (viii)* 9216 Find the prime factors of 9216: 9216 = 2 × 4608 4608 = 2 × 2304 2304 = 2 × 1152 1152 = 2 × 576 576 = 2 × 288 288 = 2 × 144 144 = 2 × 72 72 = 2 × 36 36 = 2 × 18 18 = 2 × 9 9 = 3 × 3 So, 9216 = 2^10 × 3^2 = (2^5 × 3)^2 = (32 × 3)^2 = 96^2. sqrt(9216) = sqrt(2^10) × 3^2 = 2^10/2 × 3^2/2 = 2^5 × 3^1 = 32 × 3 = 96 (ix)* 529 Find the prime factors of 529: 529 = 23 × 23 So, 529 = 23^2. sqrt(529) = sqrt(23^2) = 23 (x)* 8100 Find the prime factors of 8100: 8100 = 100 × 81 100 = 10 × 10 = (2 × 5) × (2 × 5) = 2^2 × 5^2 81 = 9 × 9 = (3 × 3) × (3 × 3) = 3^4 So, 8100 = 2^2 × 3^4 × 5^2 = (2 × 3^2 × 5)^2 = (2 × 9 × 5)^2 = (18 × 5)^2 = 90^2. sqrt(8100) = sqrt(2^2 × 3^4 × 5^2) = 2^2/2 × 3^4/2 × 5^2/2 = 2^1 × 3^2 × 5^1 = 2 × 9 × 5 = 90 Send me the next one 📸

Accepted Answer · 2026-04-16T13:33:15.601Z

You're on a roll — Step 1: Find the square root of each number using the Prime Factorisation Method. (i)* 729 Find the prime factors of 729: 729 = 3 × 243 243 = 3 × 81 81 = 3 × 27 27 = 3 × 9 9 = 3 × 3 So, 729 = 3 × 3 × 3 × 3 × 3 × 3 = (3 × 3 × 3)^2 = 27^2. sqrt(729) = sqrt(3^6) = 3^6/2 = 3^3 = 27 (ii)* 400 Find the prime factors of 400: 400 = 2 × 200 200 = 2 × 100 100 = 2 × 50 50 = 2 × 25 25 = 5 × 5 So, 400 = 2 × 2 × 2 × 2 × 5 × 5 = (2 × 2 × 5)^2 = 20^2. sqrt(400) = sqrt(2^4 × 5^2) = 2^4/2 × 5^2/2 = 2^2 × 5^1 = 4 × 5 = 20 (iii)* 1764 Find the prime factors of 1764: 1764 = 2 × 882 882 = 2 × 441 441 = 3 × 147 147 = 3 × 49 49 = 7 × 7 So, 1764 = 2 × 2 × 3 × 3 × 7 × 7 = (2 × 3 × 7)^2 = 42^2. sqrt(1764) = sqrt(2^2 × 3^2 × 7^2) = 2^1 × 3^1 × 7^1 = 2 × 3 × 7 = 42 (iv)* 4096 Find the prime factors of 4096: 4096 = 2 × 2048 2048 = 2 × 1024 1024 = 2 × 512 512 = 2 × 256 256 = 2 × 128 128 = 2 × 64 64 = 2 × 32 32 = 2 × 16 16 = 2 × 8 8 = 2 × 4 4 = 2 × 2 So, 4096 = 2^12 = (2^6)^2 = 64^2. sqrt(4096) = sqrt(2^12) = 2^12/2 = 2^6 = 64 (v)* 7744 Find the prime factors of 7744: 7744 = 2 × 3872 3872 = 2 × 1936 1936 = 2 × 968 968 = 2 × 484 484 = 2 × 242 242 = 2 × 121 121 = 11 × 11 So, 7744 = 2 × 2 × 2 × 2 × 2 × 2 × 11 × 11 = (2 × 2 × 2 × 11)^2 = (8 × 11)^2 = 88^2. sqrt(7744) = sqrt(2^6 × 11^2) = 2^6/2 × 11^2/2 = 2^3 × 11^1 = 8 × 11 = 88 (vi)* 9604 Find the prime factors of 9604: 9604 = 2 × 4802 4802 = 2 × 2401 2401 = 7 × 343 343 = 7 × 49 49 = 7 × 7 So, 9604 = 2 × 2 × 7 × 7 × 7 × 7 = (2 × 7 × 7)^2 = (2 × 49)^2 = 98^2. sqrt(9604) = sqrt(2^2 × 7^4) = 2^2/2 × 7^4/2 = 2^1 × 7^2 = 2 × 49 = 98 (vii)* 5929 Find the prime factors of 5929: 5929 = 7 × 847 847 = 7 × 121 121 = 11 × 11 So, 5929 = 7 × 7 × 11 × 11 = (7 × 11)^2 = 77^2. sqrt(5929) = sqrt(7^2 × 11^2) = 7^1 × 11^1 = 7 × 11 = 77 (viii)* 9216 Find the prime factors of 9216: 9216 = 2 × 4608 4608 = 2 × 2304 2304 = 2 × 1152 1152 = 2 × 576 576 = 2 × 288 288 = 2 × 144 144 = 2 × 72 72 = 2 × 36 36 = 2 × 18 18 = 2 × 9 9 = 3 × 3 So, 9216 = 2^10 × 3^2 = (2^5 × 3)^2 = (32 × 3)^2 = 96^2. sqrt(9216) = sqrt(2^10) × 3^2 = 2^10/2 × 3^2/2 = 2^5 × 3^1 = 32 × 3 = 96 (ix)* 529 Find the prime factors of 529: 529 = 23 × 23 So, 529 = 23^2. sqrt(529) = sqrt(23^2) = 23 (x)* 8100 Find the prime factors of 8100: 8100 = 100 × 81 100 = 10 × 10 = (2 × 5) × (2 × 5) = 2^2 × 5^2 81 = 9 × 9 = (3 × 3) × (3 × 3) = 3^4 So, 8100 = 2^2 × 3^4 × 5^2 = (2 × 3^2 × 5)^2 = (2 × 9 × 5)^2 = (18 × 5)^2 = 90^2. sqrt(8100) = sqrt(2^2 × 3^4 × 5^2) = 2^2/2 × 3^4/2 × 5^2/2 = 2^1 × 3^2 × 5^1 = 2 × 9 × 5 = 90 Send me the next one 📸

Find the square root of the following numbers using the Prime Factorisation Method: (i) 729 (ii) 400 (iii) 1764 (iv) 4096

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Find the square root of the following numbers using the Prime Factorisation Method: (i) 729 (ii) 400 (iii) 1764 (iv) 4096

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